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\nopagenumbers
\rightline{\sevenbf 4/15/99}
\bigskip
\centerline{\largerbold Review \raise .015em\hbox{\largerqq advice}
for the final exam in Math 135}
\centerline{\largerbold (including some review problems)}
\medskip
These review problems mostly cover the last few weeks of the
course. The final is cumulative and will cover all parts of the
course. Material which has not yet been tested will be emphasized more
on the final exam. Keep in mind that if we thought it was important
earlier, we probably still think it is important. Here are other
useful references.
\medskip
\item{$\bullet$} The earlier review sheets (with answers available
on the web).
%\smallskip
\vskip .035in
\item{$\bullet$} The exams {\it you}\/ had earlier in this course.
\vskip .035in
%\smallskip
\item{$\bullet$} Two real exams with answers given this semester to
one class (with answers and grading guide available on the web).
\vskip .035in
%\smallskip
\item{$\bullet$} {\bf Three real final exams} given during the last
two academic years (available on the web).
\smallskip
\noindent Math 135 material on the web including answers to
review questions can be accessed through:
\smallskip
\centerline{\tt
http://www.math.rutgers.edu/math135.html}
\smallskip
\noindent The final exam will be given {\bf Wednesday, May 5, from 4
to 7 PM} with locations to be announced separately for each
class. These review problems are intended to represent the type of
problems which {\it may}\/ appear on exams. Many routine review
problems are available in the review sections of the text.
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\centerline{\vrule height 0pt depth 1pt width 3in}
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\smallskip
Graphing calculators may be used on the Math 135 final exam but
calculators and computers with QWERTY keyboards or symbolic
differentiation and integration programs are not allowed. Students may
bring one sheet of paper containing notes (using both sides!) up to
8.5 by 11 inches in size to any exam.
\medskip
0.~Suppose $f(x)=Ce^{Dx}$. Find values of $C$ and $D$ so that $f(1)=3$
and $f(2)=4$. Sketch a graph of $y=f(x)$ for $x$ between $-2$ and $6$.
\medskip
\hskip 4.7in\vbox{
\def\tablerule{\noalign{\hrule depth0pt height.5pt width1.57in}}
\halign to \hsize{\hfil $ # $\strut\ \ \vrule\thinspace
& \hfil $ # $\hfil &\vrule\thinspace \ $ # $ \thinspace
&\vrule\thinspace \ $ # $ \thinspace \cr
\hfil\ x \hfil&\ Q(x)\thinspace\thinspace\vphantom{j}
\hfil& Q'(x)& Q''(x)\cr
\tablerule
\tablerule
0 & 1 & -4 & -3 \cr
\tablerule
1& 2 & 0 & 7 \cr
\tablerule
2 & 0 & -5 & 2 \cr
\tablerule
3 & 1 & 0 & -2 \cr
\tablerule
4 & 5 & 5 & -2 \cr
\tablerule
5 & 8 & -1 & 0 \cr
\tablerule
6 & -2 & 2 & 3 \cr
\tablerule
7 & -2 & 0 & 0 \cr
}}
\vskip -1in
\hbox{\hsize = 4.3in \vtop{1.~Suppose $Q$ and its first derivative
$Q'$ and its second derivative $Q''$ have the values indicated in the
accompanying table.
\medskip
Below are some disconnected pieces of the graph of $Q$. Each value of
$x$ matches exactly one picture. Find the matches.
\overfullrule=0pt
\bigskip\bigskip
\centerline{\hskip 2.2in\epsfxsize=6.5in\epsfbox{revf-2.eps}}
}}
\medskip
2.~Graph $y=(2x+1)e^{-x^2}$. Use calculus to give information as
precisely as possible about local extrema, intervals of increase and
decrease, inflection points, and intervals of concavity.
\medskip
3.~The line $y=3x+7$ is tangent to the graph of $y=f(x)$ when
$x=4$. What is $f(4)$? what is $f'(4)$?
\medskip
4.~Suppose $y$ is defined implicitly as a function of $x$ by
$x^2+Axy^2+By^3=1$ where $A$ and $B$ are constants to be
determined. Given that this curve passes through the point $(3,2)$ and
that its tangent at this point has slope $-1$, find $A$ and $B$.
\medskip
5.~Evaluate these indefinite integrals.
\medskip
\line{
a) $\d \int 7x^2-3e^x +{5\over x}\, dx$\qquad
b) $\d \int (x^3 +5)^2 \, dx$\qquad
c) $\d \int 5 \sin x + \cos (5x) \, dx$\qquad
d) $\d \int {x\over {x^2+5}} \, dx$\hfil
}
\vfil
\rightline{\sevenbf OVER}
\eject
6.~Find a solution to the differential equation $y'=2x^3-1$ passing
through the point $(2,3)$.
\bigskip
7.~What is the maximum value of the function $K(x)$ if $K''(x)=-9$ for
all $x$ and if also $K(0)=0$ and $K(2)=0$?
\bigskip
8.~$\d{\int_0^1 {4\over{1+x^2}}\,dx}$ and $\d{\int_0^{\sqrt{3}}
{3\over{1+x^2}}\,dx}$ are both exactly equal to $\pi$\footnote*{\rm
Really! This is true!}. Use this together with simple properties of
definite integrals to get the exact value of $\d\int_1^{\sqrt{3}}
{1\over{1+x^2}}\, dx$.
\bigskip
9.~Evaluate these definite integrals using methods of calculus.
\medskip
\line{
a) $\d{\int_1^2 \left(x^3 -{1\over{x^4}}\right) dx}$ \hfil
b) $\d{\int_0^{\ln 3} 4e^{2x} \, dx}$ \hfil
c) $\d{\int_0^{2} x^2 \sqrt{1+3x^3} \, dx}$ \hfil
d) $\d{\int_{\ln \pi}^{\ln 2\pi} e^x \sin(e^x) \, dx}$ \hfil
e) $\d{\int_e^{e^e} {1\over {x\ln x}}\, dx}$\hfil}
\bigskip
10.~a) It's known that $\d{\int x^2 e^x \, dx = Ex^2 e^x +Fxe^x +G
e^x \ (+\ {\rm any\ constant})}$ for certain specific numbers $E$ and
$F$ and $G$. What are these specific numbers $E$ and $F$ and $G$?
\medskip
b) Use your answer to a) to compute $\d \int_0^1 x^2 e^x \, dx$.
\bigskip
11.~If $F(x)=\d\int_{-42}^x {{\sin (t^2)}\over {1+t^4}}\,dt$, compute
$F(-42)$, $F'(0)$, and $F'(\sqrt{\pi})$ exactly.
\bigskip
12.~Below is a graph of $y=f(t)$.
\bigskip
\centerline{\epsfbox{revf-1.eps}}
\smallskip
\centerline{Graph of $y=f(t)$}
\medskip
Each segment of the graph is a semicircle.
Suppose that $\d{g(x)=\int_{-1}^x f(t)\; dt}$.
\medskip
a) What are $g(-1)$ and $g(0)$ and $g(5)$? What are $g'(-1)$ and
$g'(0)$ and $g'(5)$?
\medskip
b) Compute a Riemann sum estimate for $g(3)=\d\int_{-1}^3 f(t)\; dt$ associated with the
partition $-1 < 0 < 1 < 3$ where the sample points are the left
endpoints of the subintervals.
\bigskip
13.~What is the area enclosed by $y=x^5$, the $x-$axis, and the lines
$x=1$ and $x=2$?
\bigskip
14.~First, find the area enclosed by $y=\sin x$, the $x$-axis, and the
line $y=\d{\pi\over 2}$. Then find a vertical line which splits this
area into two equal parts.
\vfill\eject\end